A296920 -id:A296920 - cp's OEIS frontend

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

3, 4, 5, 11, 23, 31, 37, 47, 49, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 169, 179, 181, 191, 199, 223, 229, 251, 257, 269, 289, 311, 313, 317, 331, 353, 361, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-11))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.
For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011582, A056874, A296920, A191060.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.
The total number of elements with norm n is given by A028609.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), this sequence (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A296925 Inert rational primes in the field Q(sqrt(-10)).

Original entry on oeis.org

3, 17, 29, 31, 43, 61, 67, 71, 73, 79, 83, 97, 101, 107, 109, 113, 137, 149, 151, 163, 181, 191, 193, 199, 227, 229, 233, 239, 257, 269, 271, 283, 307, 311, 313, 337, 347, 349, 353, 359, 389, 421, 431, 433, 439, 443, 457, 461, 467, 479, 509, 523, 541, 547, 563, 577, 587, 593, 599, 617, 631, 643, 661, 673, 683, 701, 709

Offset: 1

N. J. A. Sloane, Dec 26 2017

Primes that are congruent to 3, 17, 21, 27, 29, 31, 33, or 39 mod 40. - Amiram Eldar, Nov 17 2023

Primes p such that the Legendre symbol (-10/p) = -1, i.e., -10 is not a square modulo p. - Jianing Song, Oct 23 2024

Cf. A155488, A293859, A296920.

Load the Maple program HH given in A296920. Then run HH(-10, 200); This produces A155488, A296925, A293859. - N. J. A. Sloane, Dec 26 2017

Select[Prime[Range[127]], KroneckerSymbol[-10, #] == -1 &] (* Amiram Eldar, Nov 17 2023 *)

A105885 Primes for which -13 is a primitive root.

Original entry on oeis.org

2, 3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 107, 109, 127, 131, 137, 139, 149, 179, 191, 197, 199, 241, 251, 263, 281, 283, 293, 317, 347, 349, 353, 367, 397, 401, 439, 449, 457, 467, 491, 503, 509, 523, 541, 557, 563, 571, 577, 593, 599, 607, 613, 617, 647, 659, 719, 727

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

Primes p such that Legendre(-13,p) = -1. - N. J. A. Sloane, Dec 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885. - N. J. A. Sloane, Dec 26 2017

pr=-13; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

A296926 Rational primes that decompose in the field Q(sqrt(-13)).

Original entry on oeis.org

7, 11, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631, 641, 643, 653, 673, 677, 683, 691

Offset: 1

N. J. A. Sloane, Dec 26 2017

In general, primes that decompose in Q(sqrt(-p prime)) are congruent modulo 4p to t(-1)^[t^(phi(p)/2) mod p = 1 XOR t mod min(e,4) = 1], where t are the totatives of 2p, e is the even part of phi(p), and [P] returns 1 if P else 0. In other words, if phi(p) is at least twice even, then the t are signed so that the quadratic residuosity of t mod p aligns with the congruence of +-t mod 4 to 1--the modulus 4p is thence irreducible--; if only once, then the signature simply indicates quadratic residues modulo p. The imbalance of signs in either flank (t < p, t > p) of the signature also gives the class number of Q(sqrt(-p)), up to an excess factor of 3 if p == 3 (mod 8) but != 3. [E.g., for p = 13 we have +--+++ or +++--+, so the class number of Q(sqrt(-13)) = 2; for p = 11 == 3 (mod 8) we have +++-+ or -+---, so the class number of Q(sqrt(-11)) = 3/3 = 1.] - Travis Scott, Jan 05 2023

Cf. A105885, A296920, A296927, A296928.

Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.

Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* Amiram Eldar, Nov 17 2023 *)

list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(-13,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018

Primes == {1, 7, 9, 11, 15, 17, 19, 25, 29, 31, 47, 49} (mod 52). - Travis Scott, Jan 05 2023

A296930 Inert rational primes in the field Q(sqrt(-17)).

Original entry on oeis.org

5, 19, 29, 37, 41, 43, 47, 59, 61, 67, 73, 83, 97, 103, 109, 113, 127, 151, 173, 179, 181, 191, 193, 197, 223, 233, 239, 241, 251, 263, 269, 271, 277, 307, 313, 317, 331, 337, 359, 383, 397, 401, 443, 449, 463, 467, 491, 521, 523, 541, 563, 587, 599, 601, 617, 631

Offset: 1

N. J. A. Sloane, Dec 26 2017

Primes that are congruent to 5, 15, 19, 29, 35, 37, 41, 43, 45, 47, 55, 57, 59, 61, 65, or 67 mod 68. - Amiram Eldar, Nov 17 2023

Cf. A296920, A296929, A296931.

Load the Maple program HH given in A296920. Then run HH(-17, 200); This produces A296929, A296930, A296931.

Select[Prime[Range[115]], KroneckerSymbol[-17, #] == -1 &] (* Amiram Eldar, Nov 17 2023 *)

A296936 Inert rational primes in the field Q(sqrt(11)).

Original entry on oeis.org

3, 13, 17, 23, 29, 31, 41, 47, 59, 61, 67, 71, 73, 101, 103, 109, 149, 163, 173, 179, 191, 193, 197, 199, 223, 233, 241, 251, 277, 281, 293, 311, 331, 337, 349, 367, 373, 379, 383, 409, 419, 443, 457, 461, 463, 467, 487, 499, 541, 557, 569, 587, 593, 599, 601

Offset: 1

N. J. A. Sloane, Dec 26 2017

Cf. A296935, A038881, A038882.

Load the Maple program HH given in A296920. Then run HH(11, 200); This produces A296935, A296936, A038881, A038882.

Select[Prime[Range[2, 120]], KroneckerSymbol[11, #] == -1 &] (* Amiram Eldar, Dec 24 2023 *)

A346722 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-11)) as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 37, 43, 49, 55, 61, 62, 64, 66, 68, 69, 71, 74, 76, 77, 79, 81, 83, 84, 86, 89, 103, 107, 121, 125, 128, 131, 133, 135, 138, 142, 145, 149, 152, 154, 156, 159, 163, 166, 173, 175, 177, 179, 197, 199, 201, 203

Offset: 1

Peter Munn, Jul 30 2021

In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-11) or z = (x+0.5) + (y+0.5)*sqrt(-11) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.)
R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example.
This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.

			The sequence is constructed in the same way as A346721, but the relevant prime is 11 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 6.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A296920, A345764.
Norms of primes in R: A341785.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346723 (D=-19), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A257362 Odd primes modulo which -163 is a square.

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 163, 167, 173, 179, 197, 199, 223, 227, 251, 263, 281, 307, 313, 347, 359, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 547, 563, 577, 593, 607, 641, 647, 653, 661, 673, 677, 691

Offset: 1

Robert Israel, Apr 20 2015

Contains A005846. The first members that are not in A005846 are 163 and 167.
Primes that divide some member of A202018.
Primes congruent to x^2 mod 163 for some x, 0 <= x <= 162.
Primes of the form x^2 + xy + 41y^2. Also, primes of the form x^2 - xy + 41y^2 with x and y nonnegative. - Jianing Song, Feb 19 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005846, A202018.

select(p -> isprime(p) and (p=163 or numtheory:-legendre(-163,p)=1), [seq(2*i+1,i=1..1000)]);
# Another Maple program is given in A296920. - N. J. A. Sloane, Dec 25 2017

Reap[For[p=3, p<1000, p = NextPrime[p], If[p==163 || KroneckerSymbol[-163, p] == 1, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 29 2019 *)

is(n)=isprime(n) && issquare(Mod(-163,n)) \\ Charles R Greathouse IV, Nov 28 2016

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 28 2016

A296927 Inert rational primes in the field Q(sqrt(-13)).

Original entry on oeis.org

3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 103, 107, 109, 127, 131, 137, 139, 149, 179, 191, 193, 197, 199, 211, 229, 241, 251, 263, 281, 283, 293, 311, 317, 347, 349, 353, 367, 397, 401, 409, 419, 421, 439, 443, 449, 457, 461, 467, 491, 503, 509, 523, 541, 547, 557, 563

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.

A296928 Primes p such that Legendre(-13,p) = 0 or 1.

Original entry on oeis.org

7, 11, 13, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Primes == 1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, or 49 (mod 52). - Robert Israel, Dec 27 2017

Robert Israel, Table of n, a(n) for n = 1..10003

Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.
select(isprime, [seq(seq(52*i+j, j=[1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, 49]),i=0..50)]); # Robert Israel, Dec 27 2017

cp's OEIS Frontend

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

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A296925 Inert rational primes in the field Q(sqrt(-10)).

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A105885 Primes for which -13 is a primitive root.

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A296926 Rational primes that decompose in the field Q(sqrt(-13)).

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A296930 Inert rational primes in the field Q(sqrt(-17)).

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A296936 Inert rational primes in the field Q(sqrt(11)).

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A257362 Odd primes modulo which -163 is a square.

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